### TL;DR:
In this "satirical" yet technically rigorous paper, ...
The paper earned Fardin the 2017 Ig Nobel Prize in Physics, you can...
The expression πάντα ῥεῖ (panta rhei, literally “all things flow”) ...
While Plato may not need an introduction, it may be helpful to prov...
Rheology is the branch of physics that studies how materials deform...
In rheology, the Trouton ratio (ηₑ/ηₛ) quantifies how much harder i...
Fardin invents the term “superfelidaphobic” by analogy with “superh...
In complex materials, the “viscosity” isn’t just a single number, i...
When Fardin introduces the Reynolds–Weissenberg number (Rw = τγ̇), ...
If you're curious, this is [what a Cat Cafe looks like in Japan](ht...
16
Rheology Bulletin, 83(2) July 2014
On the rheology of cats
M.A. Fardin
1, 2, 3, ∗
1
Universit´e de Lyon, Laboratoire de Physique,
´
Ecole Normale Sup´erieure de Lyon,
CNRS UMR 5672, 46 All´ee d’Italie, 69364 Lyon cedex 07, France.
2
The Academy of Bradylogists.
3
Member of the Extended McKinley Family (EMF).
(Dated: July 9, 2014)
In this letter I highlight some of the recent developments around the rheology of Felis catu s, with
p otential applications for other species of the felidae family. In the linear rheology regime many
factors can enter the determination of the characteristic time of cats: from surface effects to yield
stress. In the nonlinear rheology regime flow instabilities can emerge. Nonetheless, the flow rate,
which is the usual dimensional control parameter, can be hard to compute because cats are active
rheological materials.
παντα ρι! Everything flows! This famous aphorism
used to characterize Heraclitus’ thought is also the motto
of rheology. “Everything flows and nothing abides; ev-
erything gives way and nothing stays fixed.” a recipe for
insubordination actually from Simplicius and Plato. Ev-
erything flows? Well, it depends on the definition of a
flow; if sufficiently general, there is no doubt that there
are no exceptions to the rule! What is a flow? What is
a fluid? As pointed out from the start by Reiner, the
essential value of rheology is to recognize that states of
matter are a matter of time(s). The first time, is a time
of observation T . What is true today may not be true
tomorrow. Time over time, one day 49, the next 50.
Historically, the popular distinction between states of
matter has been made based on qualitative differences
in bulk prop erties. Solid is the state in which matter
maintains a fixed volume and shape; liquid is the state
in which matter maintains a fixed volume but adapts to
the shape of its container; and gas is the state in which
matter expands to occupy whatever volume is available.
Following these common sense definitions, a meta-study
untitled “Cats are liquids” was recently published on
boredpanda.com. I propose here to check if the panda’s
claim that the cats are liquid is solid, by using the tools
of modern rheology.
First of all, ‘maintains’, ‘adapts’ or ‘expands’ are verbs.
They describe actions unfolding with a characteristic
time scale τ , which we will call relaxation time. From
T and τ we can define the Deborah number as:
De ≡
τ
T
(1)
Usually T is just the duration of the experiment, but for
oscillatory flows it is the inverse of the frequency (and
thus De is analogous to a Strouhal number). The re-
laxation time τ can have a variety of origins. When one
seeks the difference between gas and liquid, ‘relaxing’ will
mean ‘expan di n g’ and so τ will be linked to the charac-
teristic rate of expansion of the material. The expansion
∗
Corresponding author ; Electronic address: marcantoine.fardin@
ens-lyon.fr
FIG. 1: (a) A cat appears as a solid material with a consis-
tent shape rotating and bouncing, like Silly Putty on short
time scales. We have De 1 because the time of observa-
tion is under a second. (b) At longer time scales, a cat flows
and fills an empty wine glass. In this case we have De 1.
In both cases, even if the samples are different, we can es-
timate the relaxation time to be in the range τ = 1 s to
1 min. (c-d) For older cats, we can also introduce a charac-
teristic time of expansion and distinguish between liquid (c)
and gaseous (d) feline states. [(a) Courtesy of http://cat-
b ounce.com, (b) http://www.dweebist.com/2009/07/kitten-
in-wine-glass/, (c) http://imgur.com/gallery/UuNSR, (d)
http://imgur.com/s7JtV ]
is a type of flow. In this case, we will say that we have
a gas if De 1. When one seeks the difference between
liquid and solid, ‘relaxing’ will mean ‘adapting’ and so τ
will be linked to the characteristic rate of adaptation of
the shape of the material to its container. The adapta-
tion of the shape of the material is a type of flow. In this
case, we will say that we have a liquid if De 1. Solids
‘maintain’ their shape and volume, i.e. they do not flow.
But solids can be deformed under stress. Note finally
that any flow is intrinsically made of deformations.
As illustrated in Fig. 1a, for De 1 a cat appears
17
Rheology Bulletin, 83 (2) July 2014
2
FIG. 2: (a) Extensional rheology of a cat before capillary break-up. (b) Cat on a superfelidaphobic substrate showing
a high contact angle. (c) Tilted jar experiment showing the yield stress of a kitten. (d) Spreading of a cat on a very
rough substrate. (e) Low affinity between cats and water surfaces. (f) Sliding cat on smooth floor. (g) Adhesion of
a cat on a vertical wall. [Courtesy of (a) facebook.com, (c) http://metro.co.uk/2011/02/18/ksyusha-the-kitten-is-cat-in-a-
jar-639735/ , (d) http://www.theyfailed.com/cats-sleep-anywhere/, (f) http://www.mirror.co.uk/news/world-news/youtube-
watch-hilarious-viral-of-two-882779, (g) http://amazinghandpaintedmurals.com/picture
gallery - page 3]
solid, whereas for De 1 it seems liquid. From these
preliminary experiments, knowing T we can estimate the
relaxation time to be in the range τ = 1 s to 1 min,
for normal cases of Felis catus. Note that the samples
used in Fig. 1a-b are relatively young. Older cats may
have a shorter relaxation time, and thus become liquid
more easily than agitated kittens, for which τ can reach
values as high as a few hours. The assumption of incom-
pressibility may also fail for older cats, which can acquire
gaseous properties like in Fig. 1c-d. In this letter, we will
tend to ignore this thixotropic behavior. There’s an old
saying in investing: even a dead cat will bounce if it is
dropped from high enough. Where, of course, the dead
cat bounce refers to a short-term recovery in a declining
trend.
Overall, the Deborah number is the dimensionless ex-
pression of the concept of linear viscoelasticity. The
greater the Deborah number, the more elastic/solid the
material; the smaller the Deborah number, the more vis-
cous/fluid it is. Thus, rheology suggests only two states
of matter: solids that deform; and fluids that flow. Both
gases and liquids flow, they are fluids, the first compress-
ible, the other incompressible. In general, both the fluid-
like and the solid-like properties of a material can be
complex, in the sense that the solid part may not be
purely elastic, and the fluid part may not be purely vis-
cous. For simple incompressible and athermal molecular
fluids, the relaxation time will simply be the viscous dis-
sipation time τ = δ
2
/ν, where δ is the thickness of the
momentum boundary layer and ν is the kinematic vis-
cosity. For more complex fluids, τ can have a large range
of origins, which often require chemistry and/or biology
to b e well understood.
In the first part of this letter I wish to highlight the
potential factors that have to be taken into account in
computing the value of τ for cat s. Fig. 2a shows the cap-
illary bridge formed during extensional rheometry of Felis
catus. First, in the introduction, we assumed τ tobea
scalar, but it can have a higher dimensionality. Usually
the time scale is considered as a contribution to viscos-
ity, which in the most general case is a tensor of rank
2. For simple incompressible fluids symmetry considera-
tions reduce this tensor to a scalar. The extensional vis-
cosity is si mp l y 3 times the shear viscosity. For complex
fluids, the extensional viscosity can be or d ers of mag-
nitude different, usually larger than the shear viscosity
for polymeric materials. For cats, the determination of
the Trouton ratio is complicated but the situation seems
opposite. In the absence of reliable extensional rheol-
ogy data, we can only point to the fact that when cats
are deformed along their principal axis, they tend to re-
lax more easily, suggesting that the extensional time is
smaller than the shear time. Transient strain-hardening
can nonetheless occur. Second, because, flows of cats are
usually free surface flows, the surface tension between the
cat and its surrounding medium can be important and
even dominant in the rheology, especially in CATBER
(Capillary thinning and breakup extensional rheometer)
experiments. The catpillar y number becomes important
τ = f(Ca), with Ca ≡ ηU/γ
LV
, where η is the shear
viscosity, U is a characteristic flow velo city and γ
LV
is
the surface tension (not to be confused with the defor-
mation). Let us recall that even water droplets bouncing
on hydrophobic substrates can behave elastically, with a
response time τ =
ρR
3
0
/γ, where ρ is the density and
R
0
the size of the drops. When the fluid is complex, the
situation can be even more entangled.
The wetting and general tribology of cats has not pro-
gressed enough to give a definitive answer to the capillary
dependence of the feline relaxation time. Fig. 2b gives
an example of a lotus effect of Felis catus, suggesti ng
that the substrate is superfelidaphobic. This behavior is
usually distinguished from the yield stress that cats can
also display, as shown in Fig. 2c, where the kitten cannot
flow because it is b elow its yield stress, like ketchup in
its bottle. It is still unclear what physical and chemical
properties generate superfelidaphobicity, but a Cassie-
Baxter-like model seems plausible. Here, the roughness
of the cat’s fur would be as determinant as the roughness
of the substrate, but probably with somewhat opposite
effects. Indeed, cats are oft en found to spread on rough
substrates as seen in Fig. 2d, but they have low affinity
for substrates that smooth their fur, like water in Fig. 2e.
Significant wall slip and shear localizat i on can also be in-
volved in some experiments, like shown in Fig. 2f, where
there is a very significant relative velocity b etween the
substrate and the cat. Counter-intuitively, gravity seems
(continues page 30)
30
Rheology Bulletin, 83(2) July 2014
3
FIG. 3: (a) A cat spontaneously rotates in a cylindrical jar.
(b) Normal forces and Weissenberg effect in a young sam-
ple of Felis catus. [Courtesy of (a) http://guremike.jp/, (b)
http://buzzlamp.com/10-weird-places-cats-get-stuck-in/]
to enhance adhesiveness, as shown in Fig. 2g.
In the last part of this letter, I wish to discuss the possi-
bility of flow instabilities in Felis catus. Linear viscoelas-
ticity conceptualizes the fact that if its Deborah number
is small a material is flowing. The physics of flow instabil-
ities warns us that, as the characteristic rate of deforma-
tion ˙γ increases, non trivial secondary flows emerge and
eventually become chaotic. Here, the important dimen-
sionless number will be the Reynolds-Weissenberg num-
ber (a sort of P´eclet number):
Rw ≡ τ ˙γ (2)
The limit Rw 1 defines the laminar base flow. Non-
trivial secondary flows will usually app ear around Rw ∼
1. Finally, the flow will b e turbul ent if Rw 1. For sim-
ple fluids, the relaxation time is the viscous dissipation
time, the driving force of instability is inertia and the
dimensionless number is just the usual Reynolds num-
ber Rw = Re. For more complex fluids in creeping flow
(Re = 0) recent progress on instabilities in viscoelastic
polymers and micelles solutions suggests that the rele-
vant dimensionless number is the Weissenberg number
alone, i.e. Rw = Wi if Re = 0. In this case elastic
turbulence can be achieved without inertia. We speak of
visco elastic flow instabilities.
When taken in its philosophical form, “panta rhei” is
the theory of motion: the belief that everything is dy-
namic and that the state of rest is illusory. But for cen-
turies, this ontology was superseded by Aristotle’s view-
point. He posited that in the absence of an external mo-
tive power all objects would come to rest and that moving
objects only continue to move so long as there is a power
inducing them to do so. Modern physics started when
Galileo and his followers put an end to Aristotle’s dogma
by showing that, unless acted upon by a net unbalanced
force, an object will maintain a constant velocity. This
was key to the realization that motion is relative and
preceded by the more fundamental concept of frame of
reference, e.g. the train moves with respect to the frame
of the platform, but the platform moves with respect to
the frame of the train. Note that even if rheologists have
taken Heraclitus’ doctrine as their motto, they depart
from his thoughts by a paradoxical but useful conception
of motion or flow, alternatively faithful to Aristotle or
Galileo.
Simple fluids like water are “passive”, they continue
to move or deform so long as there is a power inducing
them to do so. In this case, the typical flow rate ˙γ is sim-
ply imposed by the operator and Rw is a natural control
parameter. For cats, assuming we have a well-defined
relaxation time τ, computing Rw is still challenging be-
cause defining ˙γ can be difficult since cats are “active”
materials. They have t h eir own motive power. Like other
biologically active materials (acto-myosin gels, bacterial
swimmers, epithelium, packs, flocks, schools, etc.), they
can exhibit spontaneous rotation as shown in Fig. 3a.
Despite these difficulties, the question remains: are
cats prone to flow instabilities when Rw increases? In
a cylindrical flow geometry, instabilities in the purely in-
ertial case (i.e. Rw = Re) and in the purely elastic case
(i.e. Rw = Wi) lead to vortex flows. In the inertial
case, the centrifugal force drives this instability and is
also responsible for the deformation of the top free sur-
face, which climbs up the outer walls of the cylinder. In
the purely elastic case, the mechanism is opposite: cen-
tripetal n or mal forces (“ho op stresses”) drive the insta-
bility and are also responsible for the Weissenberg effect,
where the fluid climbs at the center of the free surface.
In general, both inertial and elastic effects can occur. In
flows of Felis catus, significant normal forces can occur
and they seem to be able to drive a Weissenberg-type
effect, as shown in Fig. 3b.
In conclusion, much more work remains ahead, but
cats are proving to be a rich model system for rheolog-
ical research, both in the linear and nonlinear regimes.
Standing questions include the potential implications of
the rheology of cats on their righting reflex, and whether
the nonlinear self-sustaining mechanism for turbulence
in pipe is applicable to streaks of tigers. Very recent ex-
periments from Japan also suggest that we should not
see cats as isolat ed fluid systems, bu t as able to transfer
and absorb stresses from their environment. Indeed, in
Japan, they have cat cafes, where stressed out customers
can pet kitties and purr their worries away.
Acknowledgments
No animals were (h)armed in the making of this
study. I thank L. and J.F. Berret for providing a
reliable technique to load Felis catus in different ge-
ometries: 1. Bring an empty b ox; 2. Wait. Il-
lustrations of the protocol can be found on the web
(weknowmemes.com/2012/11/how-to-catch-a-cat/, or
youtube.com/watch?v=pX4yK4yG3pE ). This paper was
written in honor of Gareth’s 50th birthday and his Bing-
ham award. Many members of the EMF helped moti-
vating this study, all good things in it can be found in
Gareth’s 200+ publications, and all mistakes are mine.
Help and encouragements from A. Jaish ankar, S. Man-
neville, V. Sharma and N. Taberlet were particularly
valuable.
(Cats, continued from page 17)
In rheology, the Trouton ratio (ηₑ/ηₛ) quantifies how much harder it is to stretch a fluid than to shear it (≈3 for simple liquids).
Fardin humorously notes that for cats the opposite seems true: cats elongate effortlessly (low extensional resistance) but resist lateral motion, suggesting a Trouton ratio < 1.
Fardin invents the term “superfelidaphobic” by analogy with “superhydrophobic” surfaces that repel water (lotus effect). A superfelidaphobic surface would repel cats :)
The expression πάντα ῥεῖ (panta rhei, literally “all things flow”) is attributed to the Greek philosopher Heraclitus of Ephesus (c. 535–475 BCE).
Heraclitus’ central doctrine was that change is the fundamental nature of reality. He used vivid imagery especially the river analogy:
"You cannot step twice into the same river, for fresh waters are ever flowing in upon you"
Rheology is the branch of physics that studies how materials deform and flow under applied forces.
It connects the behavior of solids (which resist deformation) and fluids (which flow)
While Plato is well known and often studied, Simplicius may need an introduction. https://plato.stanford.edu/entries/simplicius/
While Plato is well known and often studied, Simplicius may need an introduction. https://plato.stanford.edu/entries/simplicius/
While Plato is well known and often studied, Simplicius may need an introduction. https://plato.stanford.edu/entries/simplicius/
While Plato is well known and often studied, Simplicius may need an introduction. https://plato.stanford.edu/entries/simplicius/
While Plato may not need an introduction, it may be helpful to provide Simplicius with an intro:
https://plato.stanford.edu/entries/simplicius/
In complex materials, the “viscosity” isn’t just a single number, it can depend on direction. That’s why the author says viscosity can be a tensor of rank 2 (i.e., a matrix relating stress and strain in different directions).
For a simple, isotropic fluid, symmetry simplifies everything: it behaves the same in every direction, so viscosity (and τ) can be treated as a scalar (just one number).
While Plato may not need an introduction, it may be helpful to provide Simplicius with an intro:
https://plato.stanford.edu/entries/simplicius/
When Fardin introduces the Reynolds–Weissenberg number (Rw = τγ̇), he’s mixing two classical instability criteria: the Reynolds number (inertial turbulence) and the Weissenberg number (elastic turbulence).
In simple fluids, flow becomes turbulent only when inertia dominates; in complex fluids (or cats) turbulence can arise purely from elastic stresses.
The humor here is that he’s treating the famous “cat zoomies” as non-linear flow instabilities. For low Rw (calm cat), flow is laminar; around Rw ≈ 1, small perturbations appear (tail twitches, rotational modes); at Rw ≫ 1, full elastic turbulence ensues!
The paper earned Fardin the 2017 Ig Nobel Prize in Physics, you can watch the ceremony and talk [here](https://www.youtube.com/watch?v=yNwLfRpNHhI).
If you're curious, this is [what a Cat Cafe looks like in Japan](https://www.youtube.com/watch?v=zadxEvylsw8).
### TL;DR:
In this "satirical" yet technically rigorous paper, physicist Marc-Antoine Fardin applies the equations of rheology (the study of how materials deform and flow) to cats. Using the Deborah number (De = τ / T), he argues that whether a cat behaves like a solid or a liquid depends on the timescale of observation: a cat bouncing on a table (short T) acts like an elastic solid, but one melting into a container (long T) flows like a viscous fluid.
The author extends the joke with real rheological terminology - yield stress, capillary number, Weissenberg effect - to “model” feline behavior as if it were a complex fluid :)
The paper earned Fardin the 2017 Ig Nobel Prize in Physics.
